e_323011
plain-language theorem explainer
For the Fin-4 multi-index (3,2,3,0,1,1), the folded midpoint M2 numerator equals eight times the explicit integer kernel Z. Gravity analysts certifying the exact 4D Regge midpoint M2=TT identity cite this as one decided cell in chunk 14 of the kernel table. Proof is a single native decide on concrete Int arithmetic.
Claim. For indices $a=3$, $b=2$, $c=3$, $d=0$, $i=1$, $j=1$ with each index in $\{0,1,2,3\}$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8\cdot Z(a,b,c,d,i,j)$, where $Z$ is the explicit integer kernel on the same six indices.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared pointwise. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at $(a,b,c,d,i,j)$. The companion $Z$ is an explicit case-table of small integers (entries such as $4$, $-2$, and so on) on the same index domain.
The ambient module is chunk 14 of a partitioned decide sweep whose global claim is $m_2^{\mathrm{num}}=8\cdot Z$ at every cell. Upstream, both kernels live in the shared KernelCert module; downstream assembly will reassemble the cells into a single universal equality.
The local theoretical setting is pure finite enumeration: no continuum limit, no floating-point geometry, only exact Int identities on $4^6$ index tuples, discharged in manageable chunks.
proof idea
One-line wrapper: decide. Both sides reduce, under the definitions of the folded numerator and the explicit kernel table, to concrete integers; the decision procedure checks equality and closes the goal. No named lemmas are invoked beyond those definitional reductions.
why it matters
This cell is one brick in the universal statement that the midpoint M2 numerator equals eight times the explicit kernel on every Fin-4 six-tuple. That universal theorem is assembled by exhaustive fin_cases over the six indices and is the algebraic certificate behind the exact Regge midpoint M2=TT identity in 4D.
Within Recognition Science gravity work, the identity pins the discrete curvature/mass kernel used in the midpoint scheme; chunked decides keep the kernel table machine-checkable without a single monolithic proof term. The declaration itself is not a physics law, but a verified table entry the assembly theorem consumes.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.